A378835 -id:A378835 - cp's OEIS frontend

A378836 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to the digital root.

10, 1, 131, 474, 10233, 50844, 1001250, 5225775, 99980565, 536333508, 9998984322, 55188464010, 999994914558, 5683515922236, 100001648752524, 585428890525092, 10000105972653645, 60302140270087340, 1000004027662440330, 6207976859006478708, 100000111315410065850

Offset: 1

Stefano Spezia, Dec 09 2024

Chai Wah Wu, Table of n, a(n) for n = 1..43

Cf. A010888, A378835.

Cf. A378560, A378564, A378837, A378838.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; a[n_]:=If[n==1,10,Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[Median[IntegerDigits[k]]==A010888[k], c++]]; c]]; Array[a, 6]

from math import prod, factorial
from collections import Counter
from sympy.utilities.iterables import partitions
def A378836(n):
    if n==1: return 10
    c, f = 0, factorial(n-1)
    for i in range(1,9*n+1):
        for s,p in partitions(i,m=n,k=9,size=True):
            a = sorted(list(Counter(p).elements())+[0]*(n-s))
            b = a[len(a)-1>>1]+a[len(a)>>1]
            if b&1^1 and b>>1 == 1+(i-1)%9:
                v = list(p.values())
                p = prod((factorial(i) for i in v))*factorial(n-s)
                c += sum(f*i//p for i in v)
    return c # Chai Wah Wu, Dec 12 2024

a(11)-a(21) from Chai Wah Wu, Dec 12 2024

A378837 Numbers with the arithmetic mean of the digits equal to the digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 99, 999, 1029, 1038, 1047, 1056, 1065, 1074, 1083, 1092, 1119, 1128, 1137, 1146, 1155, 1164, 1173, 1182, 1191, 1209, 1218, 1227, 1236, 1245, 1254, 1263, 1272, 1281, 1290, 1308, 1317, 1326, 1335, 1344, 1353, 1362, 1371, 1380, 1407, 1416, 1425, 1434, 1443

Offset: 1

Stefano Spezia, Dec 09 2024

			10 is not in the list because the mean of its digits is not integer.
11 is not in the list because the mean of its digits is 1 which is not equal to 2, the digital root of 11.
1029 is in the list because the mean of its digits is 3 which is equal to the digital root of 1029.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A010888, A378838.

Cf. A378560, A378564, A378835, A378836.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; Select[Range[0,1500], Mean[IntegerDigits[#]]==A010888[#] &]

A378838 a(n) is the number of n-digit nonnegative integers with the mean of the digits equal to the digital root.

Original entry on oeis.org

10, 1, 1, 748, 1, 1, 373327, 1, 1, 900000000, 1, 1, 118641180477, 1, 1, 70265700376176, 1, 1, 473609016175792282, 1, 1, 25843609164809475416, 1, 1, 15917111940073972644247, 1, 1, 319908753084273214311674685, 1, 1, 6159986083122001233681300544, 1, 1, 3860982462614939076156553701616, 1, 1

Offset: 1

Stefano Spezia, Dec 09 2024

Chai Wah Wu, Table of n, a(n) for n = 1..81

Cf. A007494, A010888, A378837.

Cf. A378560, A378564, A378835, A378836.

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; a[n_]:=If[n==1,10,Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[Mean[IntegerDigits[k]]==A010888[k], c++]]; c]]; Array[a, 6]

from math import prod, factorial
from sympy.utilities.iterables import partitions
def A378838(n):
    if n==1: return 10
    if n%3!=1: return 1
    k, c, f = n%9, 0, factorial(n-1)
    a = 3*n if k==4 or k==7 else n
    for i in range(a,9*n+1,a):
        for s,p in partitions(i,m=n,k=9,size=True):
            v = list(p.values())
            p = prod((factorial(i) for i in v))*factorial(n-s)
            c += sum(f*i//p for i in v)
    return c # Chai Wah Wu, Dec 12 2024

Conjecture: a(A007494(n)) = 1.
From Chai Wah Wu, Dec 12 2024: (Start)
The above conjecture is true, i.e. if n == 0 or 2 mod 3, then a(n) = 1.
Proof: if m is a n-digit integer with mean of digits equal to its digital root k, then sum of digits of m is k*n.
Then m mod 9 = k*n mod 9. Since the digital root is k = 1 + (m-1) mod 9, this means that k = 1 + (k*n-1) mod 9. As 1<=k<=9, for n == 2, 3, 5, 6, 8, 9 mod 9 the only k that satisfies this equation is k=9. Then the only corresponding m whose digit sum is 9*n is 10^n-1. Thus a(n) = 1.
Other results:
Theorem 1: If n == 4 or 7 mod 9, then a(n) is the number of n-digit nonnegative integers with digit sum a multiple of 3*n.
Proof: Follows from the fact that the only k that satisfies k = 1 + (k*n-1) mod 9 is k = 3, 6, or 9.
Theorem 2: If n>1 and n == 1 mod 9, then a(n) is the number of n-digit nonnegative integers with digit sum a multiple of n.
Proof: Since n>1, the digital root of an n-digit integer is > 0. The result then follows from the fact that k = 1 + (k*n-1) mod 9 is satisfied for all 1<=k<=9.
(End)

a(11)-a(36) from Chai Wah Wu, Dec 12 2024

A379180 Nonnegative integers with mode and mean of the digits equal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1012, 1021, 1102, 1111, 1120, 1201, 1210, 1223, 1232, 1322, 1335, 1353, 1447, 1474, 1533, 1559, 1595, 1744, 1955, 2011, 2024, 2042, 2101, 2110, 2123, 2132

Offset: 1

Stefano Spezia, Dec 17 2024

			1021 is a term since the mode and the mean of the digits are equal to 1.

Cf. A115353, A378560, A378835, A379181.

Select[Range[0,2200], Commonest[IntegerDigits[#]] == {Mean[IntegerDigits[#]]} &]

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A378836 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to the digital root.

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A378837 Numbers with the arithmetic mean of the digits equal to the digital root.

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A378838 a(n) is the number of n-digit nonnegative integers with the mean of the digits equal to the digital root.

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Python

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A379180 Nonnegative integers with mode and mean of the digits equal.

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Mathematica